3.1330 \(\int \frac{1}{x^4 (a+b x^6)} \, dx\)

Optimal. Leaf size=40 \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{1}{3 a x^3} \]

[Out]

-1/(3*a*x^3) - (Sqrt[b]*ArcTan[(Sqrt[b]*x^3)/Sqrt[a]])/(3*a^(3/2))

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Rubi [A]  time = 0.0187189, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 325, 205} \[ -\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^6)),x]

[Out]

-1/(3*a*x^3) - (Sqrt[b]*ArcTan[(Sqrt[b]*x^3)/Sqrt[a]])/(3*a^(3/2))

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{x^4 \left (a+b x^6\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2\right )} \, dx,x,x^3\right )\\ &=-\frac{1}{3 a x^3}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^3\right )}{3 a}\\ &=-\frac{1}{3 a x^3}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x^3}{\sqrt{a}}\right )}{3 a^{3/2}}\\ \end{align*}

Mathematica [B]  time = 0.0189585, size = 101, normalized size = 2.52 \[ \frac{\sqrt{b} x^3 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+\sqrt{b} x^3 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-\sqrt{b} x^3 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )-\sqrt{a}}{3 a^{3/2} x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(a + b*x^6)),x]

[Out]

(-Sqrt[a] + Sqrt[b]*x^3*ArcTan[(b^(1/6)*x)/a^(1/6)] + Sqrt[b]*x^3*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] - Sq
rt[b]*x^3*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])/(3*a^(3/2)*x^3)

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,a{x}^{3}}}-{\frac{b}{3\,a}\arctan \left ({b{x}^{3}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(b*x^6+a),x)

[Out]

-1/3/a/x^3-1/3*b/a/(a*b)^(1/2)*arctan(b*x^3/(a*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^6+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60211, size = 194, normalized size = 4.85 \begin{align*} \left [\frac{x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{6} - 2 \, a x^{3} \sqrt{-\frac{b}{a}} - a}{b x^{6} + a}\right ) - 2}{6 \, a x^{3}}, -\frac{x^{3} \sqrt{\frac{b}{a}} \arctan \left (x^{3} \sqrt{\frac{b}{a}}\right ) + 1}{3 \, a x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^6+a),x, algorithm="fricas")

[Out]

[1/6*(x^3*sqrt(-b/a)*log((b*x^6 - 2*a*x^3*sqrt(-b/a) - a)/(b*x^6 + a)) - 2)/(a*x^3), -1/3*(x^3*sqrt(b/a)*arcta
n(x^3*sqrt(b/a)) + 1)/(a*x^3)]

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Sympy [A]  time = 0.484106, size = 71, normalized size = 1.78 \begin{align*} \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (- \frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x^{3} \right )}}{6} - \frac{\sqrt{- \frac{b}{a^{3}}} \log{\left (\frac{a^{2} \sqrt{- \frac{b}{a^{3}}}}{b} + x^{3} \right )}}{6} - \frac{1}{3 a x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(b*x**6+a),x)

[Out]

sqrt(-b/a**3)*log(-a**2*sqrt(-b/a**3)/b + x**3)/6 - sqrt(-b/a**3)*log(a**2*sqrt(-b/a**3)/b + x**3)/6 - 1/(3*a*
x**3)

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Giac [A]  time = 1.18658, size = 42, normalized size = 1.05 \begin{align*} -\frac{b \arctan \left (\frac{b x^{3}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} a} - \frac{1}{3 \, a x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(b*x^6+a),x, algorithm="giac")

[Out]

-1/3*b*arctan(b*x^3/sqrt(a*b))/(sqrt(a*b)*a) - 1/3/(a*x^3)